Let W be an inner product space and v1, . . . , vn a basis...
Let W be an inner product space and v1, . . . , vn a basis of V
. Show that
<S, T> = <Sv1, T v1> + . . . + <Svn, T vn> for S,
T ∈ L(V, W) is an inner product on L(V, W).
Let T,S : V → W be two linear transformations, and suppose B1 =
{v1,...,vn} andB2 = {w1,...,wm} are bases of V and W,
respectively.
(c) Show that the vector spaces L(V,W) and Matm×n(F) are
isomorphic. (Hint: the function MB1,B2 : L(V,W) → Matm×n(F) is
linear by (a) and (b). Show that it is a bijection. A linear
transformation is uniquely specified by its action on a basis.)
need clearly proof
5. Prove the Following:
a. Let {v1, . . . , vn} be a finite collection of vectors in a
vector space V and suppose that it is not a linearly independent
set.
i. Show that one can find a vector w ∈ {v1, . . . , vn} such
that w ∈ Span(S) for S := {v1, . . . , vn} \ {w}. Conclude that
Span(S) = Span(v1, . . . , vn).
ii. Suppose T ⊂ {v1,...
Vector v=(9,0,2) is vector from R3 space. Consider standard
inner product in R3. Let W be a subspace in R3 span by u = (9,2,0)
and w=(9/2,0,2).
a) Does V belong to W? show explanation
b) find orthonormal basis in W. Show work
c) find projection of v onto W( he best approximation of v with
elements of w)
d) find the distance between projection and vector v
In each case, check that { v1,...vn} is a basis for R^n, and
express the given vector b as a linear combination of the basis
vectors.
(a). v1=(2,3), v2=(3,5). b=(3,4)
(b) v1=(1,0,3), v2=(1,2,2), v3=(1,3,2). b=(1,1,2)
(c) v1=(1,0,1), v2=(1,1,2), v3=(1,1,1). b=(3,0,1)
Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are
vectors in W. Suppose that v1; v2 are linearly independent and that
w1;w2;w3 span W.
(a) If dimW = 3 prove that there is a vector in W that is not
equal to a linear combination of v1 and v2.
(b) If w3 is a linear combination of w1 and w2 prove that w1 and
w2 span W.
(c) If w3 is a linear combination of w1 and...
(10pt) Let V and W be a vector space over R. Show that V × W
together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1
∈W
and
λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over
R.
(5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat
(λ+μ)(u+v) = ((λu+λv)+μu)+μv.
(In your proof, carefully refer which axioms of a vector space
you use for every equality. Use brackets and refer to Axiom 2 if
and when you change them.)
V and W are finite dimensional inner product spaces,T:V→W is a
linear map, and∗represents the adjoint.
1A: Let n be a positive integer, and suppose that T is defined
on C^n (with the usual inner product) by T(z1,z2,...,zn) =
(0,z1,z2,...,zn−1). Give a formula for T*.
1B: Show that λ is an eigenvalue of T if and only if λ is an
eigenvalue of T*.